第14章流体力学

According to Bernoulli’s equation, if we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, let us consider some specific situations that simplify and illustrate its use and meaning.

First consider the very simple situation where the fluid is static—that is, $v_1=v_2=0.$ Bernoulli’s equation in that case is

We can further simplify the equation by setting $h_2=0.$ (Any height can be chosen for a reference height of zero, as is often done for other situations involving gravitational force, making all other heights relative.) In this case, we get

This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by $h_1$, and consequently, $p_2$ is greater than $p_1$ by an amount $\rho g{h}_1$. In the very simplest case, $p_1$ is zero at the top of the fluid, and we get the familiar relationship $p=\rho gh$. $(\text{Recall that}p=\rho gh$ and $\text{Δ}{U}_{\text{g}}=\text{−}mgh.)$ Thus, Bernoulli’s equation confirms the fact that the pressure change due to the weight of a fluid is $\rho gh$. Although we introduce Bernoulli’s equation for fluid motion, it includes much of what we studied for static fluids earlier.

Suppose a fluid is moving but its depth is constant—that is, $h_1=h_2$. Under this condition, Bernoulli’s equation becomes

Situations in which fluid flows at a constant depth are so common that this equation is often also called Bernoulli’s principle, which is simply Bernoulli’s equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) Bernoulli’s principle reinforces the fact that pressure drops as speed increases in a moving fluid: If $v_2$ is greater than $v_1$ in the equation, then $p_2$ must be less than $p_1$ for the equality to hold.

Many devices and situations occur in which fluid flows at a constant height and thus can be analyzed with Bernoulli’s principle.

People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment. Entrainment devices have been in use since ancient times as pumps to raise water to small heights, as is necessary for draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in Figure 14.31.